WEBVTT

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Here is the problem: Find the magnitude and unit vector of ⟨3, 4⟩.

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Vector magnitude and unit vector using exact components.

3
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Square the components and add them.

4
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Take the square root for the magnitude.

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Divide each component by the magnitude for the unit vector.

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Check: the unit vector has length 1.

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Direction angle from the positive x-axis.

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So the answer is |u| equals 5; unit vector û equals ⟨3 over 5, 4 over 5⟩; direction approximately 53.13010235°.

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Vectors are added, subtracted and scaled one component at a time. The magnitude comes from the Pythagorean theorem, and dividing by it gives a unit vector of length 1 pointing the same way.

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The unit vector prime s components square and add to 1.